where \(\lambda \) , \(\alpha _{1}\) , \(\alpha _{2}\) , \(\beta _{1}\) , \(\beta _{2}\)  are real parameters, \(\hat{a}^{\dagger } \)  and \(\hat{a}\)  are the creation and annihilation operators of the standard harmonic oscillator and \(\hat{n}=\hat{a}^\dagger \hat{a}\)  is the number operator [1]}, [2]}, [3]}. The presence of the linear terms makes the Hamiltonian in Eq. (REF ) non-\(\mathcal {PT}\) -symmetric; this can be readily demonstrated applying the parity operator \(\hat{P}:\hat{a}\rightarrow -\hat{a}\)  and the time-reversal operator \(\hat{T}: \hat{a}\rightarrow \hat{a}\) , with similar transformations for \(\hat{a}^{\dagger }\) . For the particular case \(\alpha _{1}=\alpha _{2}=0\) , we get back to the original Swanson’s quadratic model, which has a \(\mathcal {PT}\)  symmetric non-Hermitian Hamiltonian [4]}, [5]}.
